A note on the off-diagonal Muckenhoupt-Wheeden conjecture

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder'on-Zygmund operators. Namely, given $1<p<q<\mathrm{\infty }$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: $$M:L^p(v)\to L^q(u)\text{and}M:L^{q^{\mathrm{\prime }}}(u^{1-q^{\mathrm{\prime }}})\to L^{p^{\mathrm{\prime }}}(v^{1-p^{\mathrm{\prime }}}),$$ then any Calder'on-Zygmund operator $T$ and its associated truncated maximal operator $T_{\star }$ are bounded from $L^p(v)$ to $L^q(u)$. Additionally, assuming only the second estimate for $M$ then $T$ and $T_{\star }$ map continuously $L^p(v)$ into $L^{q,\mathrm{\infty }}(u)$. We also consider the case of generalized Haar shift operators and show that their off-diagonal two weight estimates are governed by the corresponding estimates for the dyadic Hardy-Littlewood maximal function.